Factorisation Problems

Introduction

Factorisation is an important idea in algebra. It helps us break down complex expressions into simpler forms. This process makes solving equations easier. It also uncovers patterns in numbers and variables. If you've ever tried to simplify expressions like 2x² + 4x or a² – b², you are already considering factorisation problems. 

In this blog, we will explore what factorisation means. We will solve common factorisation questions and provide useful formulas and tips. Whether you're studying for exams or trying to understand the topic better, these factorisation questions will help you master the skill.

 

Table of Contents

• What Is Factorisation?
• Why Learn Factorisation?
• Four Types of Factorisation
• Factorisation Formulas to Remember
• Solved Factorisation Questions
• Real-Life Applications of Factorisation
• Practice Exercises for Students
• How to Solve Factorisation Problems Effectively?
• Advanced Factorisation Example
• Conclusion
• FAQs On Factorisation Problems

 

What Is Factorisation?

Factorisation means expressing an algebraic expression as a product of its factors. Just as numbers can be broken down into prime factors (like 12 = 2 × 2 × 3), algebraic expressions can be broken into algebraic factors.

Example:

x² + 5x = x(x + 5)
Here, the expression has been written as a product of x and (x + 5). This helps in solving equations, simplifying expressions, and more.

 

Why Learn Factorisation?

Understanding factorisation is helpful for many reasons:

• It simplifies equations and helps in solving them.

• It's used in geometry, calculus, and trigonometry.

• It makes algebra manageable and efficient.

• It's useful in real-world applications like coding, data science, and economics.

The ability to factor expressions quickly is a must-have skill for students of algebra.

 

Four Types of Factorisation

There are four main types of factorisation methods that students should master.

1. Common Factors

In this type, you look for common terms in all parts of the expression.

Example:
6x + 12 = 6(x + 2)

 

2. Difference of Squares

This applies when the expression has a pattern like a² – b².

Formula:
a² – b² = (a – b)(a + b)
Example: x² – 49 = (x – 7)(x + 7)

 

3. Perfect Square Trinomials

These use identities like:

a² + 2ab + b² = (a + b)²
a² – 2ab + b² = (a – b)²

Example:
x² + 10x + 25 = (x + 5)²

 

4. Quadratic Trinomials

These are expressions like ax² + bx + c. We split the middle term and factor in pairs.

Example:
x² + 7x + 10 = (x + 2)(x + 5)

 

Factorisation Formulas to Remember

Here are some useful identities:

• a² – b² = (a – b)(a + b)

• a² + 2ab + b² = (a + b)²

• a² – 2ab + b² = (a – b)²

• a³ – b³ = (a – b)(a² + ab + b²)

• a³ + b³ = (a + b)(a² – ab + b²)

These are essential tools when solving more advanced factorisation problems.

 

Solved Factorisation Questions

Let's now go through a variety of factorisation problems that range from basic to challenging.

 

Question 1

Factorise: x² – 9

Solution:
This is a difference of squares: x² – 3²
Answer: (x – 3)(x + 3)

 

Question 2

Factorise: x² + 8x + 16

Solution:
It’s a perfect square: x² + 2×4×x + 4²
Answer: (x + 4)²

 

Question 3

Factorise: x² + 3x – 10

Solution:
Find two numbers that multiply to –10 and add to 3 → 5 and –2
Answer: (x + 5)(x – 2)

 

Question 4

Factorise: 2x² + 7x + 3

Solution:
Multiply 2 × 3 = 6
Find two numbers that multiply to 6 and add to 7 → 6 and 1
Break middle term:
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)

 

Question 5

Factorise: 3a² – 12a

Solution:
Common factor: 3a
Answer: 3a(a – 4)

 

Question 6

Factorise: 49x² – 36

Solution:
= (7x)² – (6)² → a² – b²
Answer: (7x – 6)(7x + 6)

 

Question 7

Factorise: x³ + 3x² – x – 3

Solution:
Group terms:
= (x³ + 3x²) – (x + 3)
= x²(x + 3) – 1(x + 3)
= (x + 3)(x² – 1)
= (x + 3)(x – 1)(x + 1)

 

Real-Life Applications of Factorisation

Factorisation is not just theoretical it has many real-world applications:

• Engineering: Simplifying equations in electronics and mechanics.

• Computer Science: Cryptography uses factorisation of large numbers.

• Finance: Mathematical models in economics often rely on algebraic simplifications.

• Physics: Formulas are factored to analyze motion, force, and energy.

When you solve a factorisation problem, you're building logic and problem-solving skills that apply in various fields.

 

Practice Exercises for Students

Try these additional questions:

1. Factorise: x² – 2x – 35

2. Factorise: 4x² – 9

3. Factorise: a² + ab – 6b²

4. Factorise: x³ – 27

5. Factorise: x² – 4x + 4

 

How to Solve Factorisation Problems Effectively?

Here are some tips:

• Always start by checking for a common factor.

• Look for patterns – especially squares, cubes, and trinomials.

• Use grouping to simplify four-term expressions.

• Apply formulas correctly – memorize the most common identities.

• Practice regularly to improve speed and accuracy.
  
  

Advanced Factorisation Example

Problem: What is the factorisation of

2axy² – 10x – 3ay² + 15?

Step-by-step:
Group terms:
(2axy² – 3ay²) + (–10x + 15)
= ay²(2x – 3) – 5(2x – 3)
= (2x – 3)(ay² – 5)

Answer: (2x – 3)(ay² – 5)

 

Conclusion

Factorisation is a basic idea in algebra that helps break expressions into their simple parts. When students learn to solve factorisation problems, they improve their problem-solving skills, simplify complicated equations, and get ready for more advanced math.

We explored:

• What factorisation is

• Types and methods

• Common formulas

• Real-world applications

• Practice questions and answers

• FAQs for better understanding

With regular practice, anyone can become confident at solving questions of factorization.

 

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FAQs On Factorisation Problems

1.  What are 4 types of factorisation?

The main types are:

1. Common factors

2. Difference of squares

3. Perfect square trinomials

4. Quadratic trinomials

 

2. What is the formula of a³–b³?

Answer:
a³ – b³ = (a – b)(a² + ab + b²)

Also remember:
a³ + b³ = (a + b)(a² – ab + b²)

 

3. What is an example of a factorisation problem?

Example: Factorise x² – 7x + 12
Solution: (x – 3)(x – 4)

 

4. How do you solve factorisation problems?

To solve:

1. Look for common factors.

2. Identify patterns (difference of squares, trinomials).

3. Use algebraic identities.

4. Group terms if there are 4 parts.
   
   

 

5. What is the factorisation of 2axy² – 10x – 3ay² + 15?

As solved earlier:
Answer: (2x – 3)(ay² – 5)

 

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